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Intlab  Interval Laboratory
"... . INTLAB is a Matlab toolbox supporting real and complex interval scalars, vectors, and matrices, as well as sparse real and complex interval matrices. It is designed to be very fast. In fact, it is not much slower than the fastest pure floating point algorithms using the fastest compilers available ..."
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Cited by 56 (2 self)
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. INTLAB is a Matlab toolbox supporting real and complex interval scalars, vectors, and matrices, as well as sparse real and complex interval matrices. It is designed to be very fast. In fact, it is not much slower than the fastest pure floating point algorithms using the fastest compilers available (the latter, of course, without verification of the result). Portability is assured by implementing all algorithms in Matlab itself with exception of exactly three routines for switching the rounding downwards, upwards and to nearest. Timing comparisons show that the used concept achieves the anticipated speed with identical code on a variety of computers, ranging from PC's to parallel computers. INTLAB may be freely copied from our home page. 1. Introduction. The INTLAB concept splits into two parts. First, a new concept of a fast interval library is introduced. The main advantage (and difference to existing interval libraries) is that identical code can be used on a variety of computer a...
On directed interval arithmetic and its applications
, 1995
"... We discuss two closely related interval arithmetic systems: i) the system of directed (generalized) intervals studied by E. Kaucher, and ii) the system of normal intervals together with the outer and inner interval operations. A relation between the two systems becomes feasible due to introduction ..."
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Cited by 16 (4 self)
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We discuss two closely related interval arithmetic systems: i) the system of directed (generalized) intervals studied by E. Kaucher, and ii) the system of normal intervals together with the outer and inner interval operations. A relation between the two systems becomes feasible due to introduction of special notations and a socalled normal form of directed intervals. As an application, it has been shown that both interval systems can be used for the computation of tight inner and outer inclusions of ranges of functions and consequently for the development of software for automatic computation of ranges of functions.
Using Directed Acyclic Graphs to Coordinate Propagation and Search for Numerical Constraint Satisfaction Problems
 In Proceedings of the 16th IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2004
, 2004
"... A. NEUMAIER [1] has given the fundamentals of interval analysis on directed acyclic graphs (DAGs) for global optimization and constraint propagation. We show in this paper how constraint propagation on DAGs can be made efficient and practical by: (i) working on partial DAG representations; and (ii) ..."
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Cited by 14 (5 self)
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A. NEUMAIER [1] has given the fundamentals of interval analysis on directed acyclic graphs (DAGs) for global optimization and constraint propagation. We show in this paper how constraint propagation on DAGs can be made efficient and practical by: (i) working on partial DAG representations; and (ii) enabling the flexible choice of the interval inclusion functions during propagation. We then propose a new simple algorithm which coordinates constraint propagation and exhaustive search for solving numerical constraint satisfaction problems. The experiments carried out on different problems show that the new approach outperforms previously available propagation techniques by an order of magnitude or more in speed, while being roughly the same quality w.r.t. enclosure properties. I.
A Lucid Interval
 American Scientist
, 2003
"... Give a digital computer a problem in arithmetic, and it will grind away methodically, tirelessly, at gigahertz speed, until ultimately it produces the wrong answer. The cause of this sorry situation is not that software is full of bugs—although that is very likely true as well— nor is it that hardwa ..."
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Cited by 13 (0 self)
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Give a digital computer a problem in arithmetic, and it will grind away methodically, tirelessly, at gigahertz speed, until ultimately it produces the wrong answer. The cause of this sorry situation is not that software is full of bugs—although that is very likely true as well— nor is it that hardware is unreliable. The problem is simply that computers are discrete and finite machines, and they cannot cope with some of the continuous and infinite aspects of mathematics. Even an innocentlooking number like 1 ⁄10 can cause no end of trouble: In most cases, the computer cannot even read it in or print it out exactly, much less perform exact calculations with it. Errors caused by these limitations of digital machines
Interval Computations and IntervalRelated Statistical Techniques: Tools for Estimating Uncertainty of the Results of Data Processing and Indirect Measurements
"... In many practical situations, we only know the upper bound ∆ on the (absolute value of the) measurement error ∆x, i.e., we only know that the measurement error is located on the interval [−∆, ∆]. The traditional engineering approach to such situations is to assume that ∆x is uniformly distributed on ..."
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Cited by 4 (1 self)
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In many practical situations, we only know the upper bound ∆ on the (absolute value of the) measurement error ∆x, i.e., we only know that the measurement error is located on the interval [−∆, ∆]. The traditional engineering approach to such situations is to assume that ∆x is uniformly distributed on [−∆, ∆], and to use the corresponding statistical techniques. In some situations, however, this approach underestimates the error of indirect measurements. It is therefore desirable to directly process this interval uncertainty. Such “interval computations” methods have been developed since the 1950s. In this chapter, we provide a brief overview of related algorithms, results, and remaining open problems.
Diagrammatic representation for interval arithmetic
 LINEAR ALGEBRA AND ITS APPLICATIONS
, 2001
"... The paper presents a diagrammatic representation of a standard interval space (the socalled MRdiagram), and shows how to represent and perform interval arithmetic and derive its various properties using the diagram. The representation is an extension and refinement of the ISdiagram representation ..."
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Cited by 4 (0 self)
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The paper presents a diagrammatic representation of a standard interval space (the socalled MRdiagram), and shows how to represent and perform interval arithmetic and derive its various properties using the diagram. The representation is an extension and refinement of the ISdiagram representation devised earlier by the author to represent interval relations. First, the MRdiagram is defined together with appropriate graphical notions and constructions for basic interval relations and operations. Second, diagrammatic constructions for all standard arithmetic operations are presented. Several examples of the use of these constructions to aid reasoning about various simple, though nontrivial, properties of interval arithmetic are included in order to show how the representation facilitates both deeper understanding of the subject matter and reasoning about its properties.
Combining Multiple Inclusion Representations in Numerical Constraint Propagation
 Publications Three Representative Papers
"... Abstract — This paper proposes a novel generic scheme enabling the combination of multiple inclusion representations to propagate numerical constraints. The scheme allows bringing into the constraint propagation framework the strength of inclusion techniques coming from different areas such as inter ..."
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Cited by 4 (4 self)
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Abstract — This paper proposes a novel generic scheme enabling the combination of multiple inclusion representations to propagate numerical constraints. The scheme allows bringing into the constraint propagation framework the strength of inclusion techniques coming from different areas such as interval arithmetic, affine arithmetic and mathematical programming. The scheme is based on the DAG representation of the constraint system. This enables devising finegrained combination strategies involving any factorable constraint system. The paper presents several possible combination strategies for creating practical instances of the generic scheme. The experiments reported on a particular instance using interval constraint propagation, interval arithmetic, affine arithmetic and linear programming illustrate the flexibility and efficiency of the approach. I.
Basic statistical methods for interval data
 Statistica Applicata [Italian Journal of Applied Statistics
, 2005
"... Real world data analysis is often affected by different type of errors as: measurement errors, computation errors, imprecision related to the method adopted for estimating the data (parameters). The uncertainty in the data, which is strictly connected to the above errors, may be treated by consideri ..."
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Cited by 3 (0 self)
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Real world data analysis is often affected by different type of errors as: measurement errors, computation errors, imprecision related to the method adopted for estimating the data (parameters). The uncertainty in the data, which is strictly connected to the above errors, may be treated by considering, rather than a single value for each data, the interval of values in which it may fall: the interval data. This kind of data representation imposes a new formulation of the classical statistical methods in the case that intervalvalued variables are considered. Accordingly, purpose of the present work is to develop suitable statistical methods for: obtaining a synthesis of the data, analysing the variability in the data and the existing relations among intervalvalued variables. The proposed solutions are based on the following assessments: – The developed statistics for intervalvalued variables are intervals. – Statistical methods for intervalvalued variables embrace classical statistical methods as special cases. – The proposed interval solutions do not contain redundant elements with respect to a given criterion. In the present work particular interest is devoted to the proof of the properties of the proposed techniques and to the comparison of the obtained results with those already existing in the literature.
CONSTRAINTENABLED DESIGN INFORMATION REPRESENTATION FOR MECHANICAL PRODUCTS OVER THE INTERNET
, 2003
"... This dissertation was presented by ..."